%5E-3-binomial-expansion.jpeg "(1-x)^-3 Binomial Expansion")
Binomial Expansion of (1-x)^-3
Introduction
In algebra, the binomial theorem is a fundamental concept that describes the expansion of powers of a binomial expression. One of the most important applications of the binomial theorem is in the expansion of negative integer powers of a binomial expression. In this article, we will explore the binomial expansion of (1-x)^-3.
Binomial Theorem
The binomial theorem states that for any positive integer n, the expression (a+b)^n can be expanded as:
$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
Expansion of (1-x)^-3
To expand (1-x)^-3, we can use the binomial theorem with a = 1, b = -x, and n = -3. Note that when n is a negative integer, the binomial theorem can be extended to:
$(a+b)^{-n} = \sum_{k=0}^{\infty} \binom{-n}{k} a^{-n-k} b^k$
Using this formula, we can expand (1-x)^-3 as:
$(1-x)^{-3} = \sum_{k=0}^{\infty} \binom{-3}{k} 1^{-3-k} (-x)^k$
Simplifying the Expansion
Using the formula for binomial coefficients, we can simplify the expansion as:
$(1-x)^{-3} = 1 + (-3)(-x) + \frac{(-3)(-4)}{2!} (-x)^2 + \frac{(-3)(-4)(-5)}{3!} (-x)^3 + ...$
Final Expansion
Simplifying further, we get:
$(1-x)^{-3} = 1 + 3x + 6x^2 + 10x^3 + 15x^4 + ...$
Conclusion
In this article, we have derived the binomial expansion of (1-x)^-3 using the binomial theorem. The expansion is an infinite series, and we have shown how to simplify it to obtain the final expansion.
References
- ** Algebra and Trigonometry** by Michael Sullivan
- Calculus by Michael Spivak
